 ## CS 334 Programming Languages Spring 2002 Assignment 9 Due Thursday, 5/2/2002

1. Please do problem 10.32 on page 10-46 of the text. To see what the problems are with multiple inheritance in C++, please read problem 10.16 on page 10-44. In particular, think about the ambiguities and arbitrary decisions that C++ must make in order to solve the problem stated illustrated in 10.16, and show that these ambiguities do not arrise in Java with implementation of multiple interfaces.

2. Variables can be said to have both L-values and R-values, where the L-value of a variable is the location it denotes and the R-value is the value stored in that location. The environment keeps track of the locations corresponding to variables, while the store (or state) keeps track of the values stored in locations. The semantics of the assignment statement for most programming languages is usually given as follows: When V := E is executed, the L-value of V is first obtained from the environment (call it l), then the expression E is evaluated, and that resulting value is stored in the location l. While side-effects can create great confusion in terms of what the result of an assignment statement will be; in this problem, side-effects are not the problem!

a) Describe in detail the effect of executing the Pascal assignment

```	a[a[i]] := a[i] + 1
```
b) Is it always the case (in Pascal) that the value of L after executing assignment L := E is equal to the value of E before executing the command? I.e. if you execute the following code:
```	write(E); L := E; writeln(L);
```
will the two values printed always be the same? For the purposes of this problem assume that the evaluation of E has no side effects. Hint : consider the assignment in part (a) with i = 2, a = 2, a = 0.

c) What does this tell you about the validity of the axiomatic rule for assignment,{P [E / L]} L := E {P}, (e.g., if L = a[a[i]], E = a[i] + 1, and P is a[a[i]] = 3)? Suggest a restriction which makes it valid.

3. Use the formal rules for axiomatic semantics given in the lecture notes to prove the correctness of the following program:
```{Precondition: n > 0}
i <- n
fact <- 1
while i > 0 do
{assert:  ...}
fact <- fact * i
i <- i - 1
end while
{Postcondition:	fact = 1*2*...*n}```
Hint: You need to figure out the loop invariant before you can complete the proof. Your proof should take the same form as the one in the lecture notes - hand-waving is NOT acceptable. Be sure to prove the entire algorithm is correct with respect to the precondition and postcondition. The lecture notes example only includes showing the loop invariant is correct. You must prove the entire program correct with respect to the precondition and postcondition!

Note: If you prefer to use the weakest precondition rules in the text, be my guest, but I suspect you will find it easier to use those given in class instead.

4. Do problem 13.14 on page 13-32 of the text using the "small-step" operational semantics given in the text. You may work just with an environment (no state) as the book does, or may use an environment and state as we did in class with the big step semantics. Please show every step of the reduction.

5. Do problem 13.15 on page 13-32 of the text.